Graph Topologies and Uniform Convergence in Quasi-Uniform ...

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138 j. rodríguez-lópez, s. romagueraFix U 0 ∈ U and let U, V ∈ U such that V 2 ⊆ U ⊆ U 0 . If x ∈ {a V },f V (x) = f(b V ). Furthermore, (x, a V ) ∈ V , so (x, b V ) ∈ V 2 ⊆ U.On the other hand, if x ∈ X\{a V } then f(x) ≤ f V (x) = f(x) ∨ f(b V ).Hence {f U } U∈U converges to f with respect to T (H + ). However, thisU −1 ×Vnet does not converge to f with respect to the topology of uniform convergencebecause (f(a U ), f U (a U )) = (f(a U ), f(b U )) ∉ W for all U ∈ U.Remark 1. As a particular case of the above theorem we can consider thecontinuous lattice R ∗ with the usual order where R ∗ = R ∪ {−∞, +∞}. Wecan consider the Scott topology σ(R ∗ ) of this continuous lattice and a quasiuniformityU on R ∗ such that T (U) = σ(R ∗ ). In every continuous lattice thesup operation is continuous considering the Scott topology (see [9, Chapter I,Proposition 1.11]). Therefore, we can apply the above theorem to this lattice.In particular, we can consider the extended lower quasi-pseudo-metric onR ∗ given by⎧⎪⎨ (x − y) ∨ 0 if x, y ∈ Rl ∗ (x, y) = 0 if x = −∞ or y = +∞⎪⎩+∞ if x = +∞ or y = −∞ and x ≠ y.The topology generated by this extended quasi-pseudo-metric has as abase all the sets of the form (a, +∞] = {b ∈ R ∗ : a ≪ b} where a ∈ R ∗and the open set {+∞} which coincide with the basic open sets in the Scotttopology (see [9, Chapter I, Proposition 1.10]). Now, the continuous functionsbetween X and R ∗ are the extended lower semicontinuous functions, i.e. lowersemicontinuous functions with values in R ∗ (see [3, 9]). Therefore, the aboveresult asserts that given a quasi-uniform space (X, U) then every extendedlower semicontinuous function on X is quasi-uniformly continuous if and onlyif the topology of uniform convergence agrees with the upper Hausdorff quasiuniformtopology induced by U −1 ×U l ∗ where U l ∗ denotes the quasi-uniformityinduced by l ∗ .Remark 2. We notice that the above theorem is also true if we considerfunctions from a quasi-uniform space to a lattice contained in a Scott quasiuniformsemigroup. Therefore, we not only can consider the extended realline R ∗ but the real line R. Consequently, given a quasi-uniform space (X, U)then SC(X) = UC(X, R) if and only if the topology of uniform convergencecoincides with the upper Hausdorff quasi-uniform topology induced byU −1 × U l .
graph topologies and quasi-uniform convergence 139We recall that a T 0 -space Z is called injective if every continuous mapf : X → Z extends continuously to any space Y containing X as a subspace.Corollary 1. Let (X, U) be a quasi-uniform space and (Y, τ) an injectivetopological T 0 -space. Let V be a quasi-uniformity compatible with τ. Thefollowing statements are equivalent:byi) Every continuous function from X to Y is quasi-uniformly continuous.ii) The proximal topology of (X × Y, U −1 × V) agrees with the topology ofuniform convergence on C(X, Y ).iii) The upper Hausdorff quasi-uniform topology induced by U −1 × V agreeswith the topology of uniform convergence on C(X, Y ).Proof. If Y is a T 0 space it is evident that the specialization order ≤ givenx ≤ y ⇔ x ∈ {y}is a partial order and it can be proved (see [9, Chapter II, Theorem 3.8]) thatif Y is an injective space then Y with the specialization order is a continuouslattice. Furthermore, in a continuous lattice the sup operation is jointlycontinuous with respect to the Scott topology (see [9, Chapter I, Proposition1.11]), so we can apply the theorem above, but the Scott topology (see [9,Chapter II, Theorem 3.8]) is homeomorphic to τ so we have completed theproof.AcknowledgementsThe authors are grateful to the referee, in particular for suggesting ashorter proof of Theorem 1.References[1] Baronti, M., Papini, P., Convergence of sequences of sets, in “Methodsof Functional Analysis in Approximation Theory”, ISNM 76, Birkhäuser-Verlag, 1986, 135 – 155.[2] Beer, G., Metric spaces on which continuous functions are uniformly continuousand Hausdorff distance, Proc. Amer. Math. Soc., 95 (1985), 653 – 658.[3] , “Topologies on Closed and Closed Convex Sets”, vol. 268, KluwerAcademic Publishers, Dordrecht, 1993.[4] Beer, G., Lechicki, A., Levi, S., Naimpally, S., Distance functionalsand suprema of hyperspace topologies, Ann. Mat. Pura Appl., 162 (1992),367 – 381.[5] Berthiaume, G., On quasi-uniformities in hyperspaces, Proc. Amer. Math.Soc., 66 (1977), 335 – 343.
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